List of top Mathematics Questions asked in UP Board XII

Find the shortest distance between the lines \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]

Find the minimum value of  ( z = x + 3y ) under the following constraints: 

• x + y ≤ 8
• 3x + 5y ≥ 15
• x ≥ 0, y ≥ 0

Differentiate \( y = x^x + (\cos x)^{\tan x \) with respect to \( x \).}

Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

If \( R \) is the relation "less than" from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also, define this relation from \( B \) to \( A \).

Find the value of: \[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]
Find the vector equation of the line which passes through the point \( (5,2,-4) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
Find the probability of 53 Sundays in a leap year.
Differentiate \( y = (\cos x)^{\sin x \).}
A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?
Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]
If \( y = \frac{x^2 + 3x + 4}{e^x \cos x} \), find \( \frac{dy}{dx} \).
\( R \) is a relation on a set of natural numbers \( N \) defined by \[ R = \{(a, b): a, b \in N \text{ and } a = b^2 \} \] Is \( (a, b) \in R \), \( (b, c) \in R \Rightarrow (a, c) \in R \) true? Justify it by one example.
The probability of \( A \) winning the race is \( \frac{1}{3} \) and that of \( B \) is \( \frac{1}{4} \). In this race, find the probability that neither \( A \) nor \( B \) can win the race.
Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]
If \( y = Ae^x + B \) where \( A, B \) are constants, then show that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \).
Find the principal value of \( \sin^{-1} \left( \sin \frac{7\pi}{4} \right) \).
Show that the function

\[ f(x) = \begin{cases} x^2 + 2, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \]

is not continuous at \( x = 0 \).
(e) The degree of the differential equation \( d^2y/dx^2 = \sqrt{x} + \left(\frac{dy}{dx}\right)^2 \) is:
(d) If vectors \( \mathbf{5i - \lambda j + 2k} \) and \( \mathbf{2i + 3j + 4k} \) are perpendicular, find \( \lambda \):
(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:
(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?
(a) If orders of matrices A and B are \( p \times q \) and \( q \times r \) respectively, then order of AB is:

Find the area of the bounded region of

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

In the set of real numbers, the relation \( R \) defined by \( R = \{(a, b) : a \leq b^2 \} \) is: